Existence of a nonnegative solution for a system of boundary value problems with nonlinear boundary conditions that involve multiplication

We consider system of two differential equations of arbitrary order endowed with nonlocal, nonlinear boundary conditions. Nonlinearity of the boundary conditions is caused by multiplication of derivatives at interior points. We prove the existence of a nonnegative solution by applying classical results of the fixed point index theory. Nonnegative solution here means that at least one component of the solution is positive. To illustrate the result an example is considered.